$C_{2^n}$-equivariant rational stable stems and characteristic classes

Abstract: In this short note, we compute the rational

C_{2^{n}}

-equivariant stable stems and give minimal presentations for the

R O (C_{2^{n}})

-graded Bredon cohomology of the equivariant classifying spaces

B_{C_{2^{n}}} S^{1}

and

B_{C_{2^{n}}} S i g m a_{2}

over the rational Burnside functor

A_{m a t h b f Q}

. We also examine for which compact Lie groups

L

the maximal torus inclusion

T o L

induces an isomorphism from

H_{C_{2^{n}}}^{*} (B_{C_{2^{n}}} L; A_{m a t h b f Q})

onto the fixed points of

H_{C_{2^{n}}}^{*} (B_{C_{2^{n}}} T; A_{m a t h b f Q})

under the Weyl group action. We prove that this holds for

L = U (m)

and any

n, m g e 1

but does not hold for

L = S U (2)

and

n > 1

.

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